#135480

order	512
nilpotency class	4
isoclinism	isoclinic class of #135479
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1819
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

548 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,12,14,7)(4,10,18,20)(5,9)(6,11)(8,13,24,30)(15,27,29,22)(16,23,31,21)(17,26)(19,25,32,28);(1,3,6,15)(2,7,11,22)(4,13,18,16)(5,14,17,29)(8,10,21,20)(9,12,26,27)(19,31,32,30)(23,25,24,28);(1,3)(2,8,9,23)(4,16)(5,14)(6,15)(7,20,12,28)(10,27,25,22)(11,21,26,24)(13,18)(17,29)(19,30)(31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=d,e^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fg,e^c=ei,f^c=fghi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42676);